Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation ∆S = ∆H for the first order variation of the entanglement entropy ∆S and the expectation value of the modular Hamiltonian ∆H. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation ∆S = ∆H for vacuum state tomography and obtain modified versions of the Bekenstein bound.

Abstract:We extend the approach of [9] to a new calculation of the Rényi entropy of a general CFT in d dimensions with a spherical entangling surface, in terms of certain thermal partition functions. We apply this approach to calculate the Rényi entropy in various holographic models. Our results indicate that in general, the Rényi entropy will be a complicated nonlinear function of the central charges and other parameters which characterize the CFT. We also exhibit the relation between this new thermal calculation and a conventional calculation of the Rényi entropy where a twist operator is inserted on the spherical entangling surface. The latter insight also allows us to calculate the scaling dimension of the twist operators in the holographic models.

We examine holographic entanglement entropy with higher curvature gravity in the bulk. We show that in general Wald's formula for horizon entropy does not yield the correct entanglement entropy. However, for Lovelock gravity, there is an alternate prescription which involves only the intrinsic curvature of the bulk surface. We verify that this prescription correctly reproduces the universal contribution to the entanglement entropy for CFT's in four and six dimensions. We also make further comments on gravitational theories with more general higher curvature interactions. A. Fefferman-Graham expansion 40 B. Curved boundaries 41 C. EE in the GB gravity 44 C.1 EE for a sphere with general d 45 C.2 Spherical entangling surfaces beyond GB gravity. 47 C.3 EE for a cylinder with general d 48 D. Curvature tensor for a warped geometry 50 -1 -1If the calculation is done in a Minkowski signature background, the extremal area is only a saddle point. However, if one first Wick rotates to Euclidean signature, the extremal surface will yield the minimal area. In either case, the area must be suitably regulated to produce a finite answer. Note that for a d-dimensional boundary theory, the bulk has d + 1 dimensions while the surface m has d − 1 dimensions. We are using 'area' to denote the (d − 1)-dimensional volume of m.3 Here, we follow closely the notation of [19].

We construct a new class of entanglement measures by extending the usual definition of Rényi entropy to include a chemical potential. These charged Rényi entropies measure the degree of entanglement in different charge sectors of the theory and are given by Euclidean path integrals with the insertion of a Wilson line encircling the entangling surface. We compute these entropies for a spherical entangling surface in CFT's with holographic duals, where they are related to entropies of charged black holes with hyperbolic horizons. We also compute charged Rényi entropies in free field theories.ArXiv ePrint: 1310.4180 arXiv:1310.4180v2 [hep-th] An interesting exercise to gain better intuition for this background gauge field (2.15) is to expand the coordinates near the spherical entangling surface: t E = ρ sin θ and r = R + ρ cos θ with ρ R. To leading order in ρ/R, one then finds that the potential reduces to B µ E 2π dθ.

We study the extension of the approach to the a-theorem of Komargodski and Schwimmer to quantum field theories in d = 6 spacetime dimensions. The dilaton effective action is obtained up to 6th order in derivatives. The anomaly flow a UV − a IR is the coefficient of the 6-derivative Euler anomaly term in this action. It then appears at order p 6 in the low energy limit of npoint scattering amplitudes of the dilaton for n ≥ 4. The detailed structure with the correct anomaly coefficient is confirmed by direct calculation in two examples: (i) the case of explicitly broken conformal symmetry is illustrated by the free massive scalar field, and (ii) the case of spontaneously broken conformal symmetry is demonstrated by the (2,0) theory on the Coulomb branch. In the latter example, the dilaton is a dynamical field so 4-derivative terms in the action also affect n-point amplitudes at order p 6 . The calculation in the (2,0) theory is done by analyzing an M5-brane probe in AdS 7 × S 4 . Given the confirmation in two distinct models, we attempt to use dispersion relations to prove that the anomaly flow is positive in general. Unfortunately the 4-point matrix element of the Euler anomaly is proportional to stu and vanishes for forward scattering. Thus the optical theorem cannot be applied to show positivity. Instead the anomaly flow is given by a dispersion sum rule in which the integrand does not have definite sign. It may be possible to base a proof of the a-theorem on the analyticity and unitarity properties of the 6-point function, but our preliminary study reveals some difficulties.arXiv:1205.3994v1 [hep-th]